The buildings that house the City Under Siege Exhibition date back to before 1726, and are probably among the first buildings constructed by the British on the Rock. That said, the architecture looks more Spanish than British, so they could be earlier.
As the area is known as Willis's Magazine, it seems quite probable that the two tent shaped windowless buildings were gunpowder stores, and that the central courtyard was used as an ammunitions laboratory.
The name presumably relates to the eponymous colonel of the Charles Willis's Regiment of Foot which became the 30th (Cambridgeshire) Regiment of Foot in 1751. The regiment was stationed in Gibraltar during the siege of 1727.
The actual exhibition consists of various tableaux illustrating the extreme suffering of the inhabitants of Gibraltar during the great siege. Some of these are looking a little tired and, like their companions up at the Great Siege Tunnels, may not be around for much longer.
More interesting is the graffiti scratched by bored soldiers trying to stay awake, as falling asleep on duty was an offence punishable by death in those days.